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Class 11

>>Applied Mathematics

>>Straight lines

>>Introduction

>>In the coordinate plane, the points F( -

Question

In the coordinate plane, the points F(−2,1),G(1,4) and H(4,1) lie on a circle with center P. What are the coordinates of point P ?

A

(0,0)

B

(1,1)

C

(1,2)

D

(1,−2)

Medium

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Solution

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Correct option is B)

Let H≡(x

1

,y

1

)=(4,1)

G≡(x

2

,y

2

)=(1,4)

F≡(x

3

,y

3

)=(−2,1)

P≡(x,y)

slope of m

r

=

x

2

−x

1

y

2

−y

1

=

1−4

4−1

=−1

slope of m

t

=

x

3

−x

2

y

3

−y

2

=

−2−1

1−4

=1

Formula for x-coordinate of the center of the circle is given as:

x=

2(m

r

−m

t

)

m

r

m

t

(y

3

−y

1

)+m

r

(x

2

+x

3

)−m

t

(x

1

+x

2

)

∴x=

2(−1−1)

(−1)(1)(1−1)+−1(1+(−2))−1(4+1)

∴x=

−4

0+1−5

=1

y=−

m

r

1

[x−

2

x

1

+x

2

]+

2

y

1

+y

2

∴y=−

−1

1

[1−

2

4+1

]+

2

1+4

∴y=[

2

2−5

]+

2

5

=1

∴P≡(1,1)

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